Application of immersed boundary methods to non-Newtonian yield-pseudoplastic flows

The interaction force between fluid and particles in particulate flows is highly dependent on the rheology of fluid. In some engineering applications, like mineral processing, the rheological behaviour of the carrier fluid is characterized by a yield-pseudoplastic non-Newtonian model. The immersed boundary method (IBM) is a numerical strategy to simulate the interaction between phases in the particulate systems using a Cartesian fixed grid. It imposes the solid boundary condition in the computational domain through a modification (e.g. a forcing term) in the governing equations. However, there is still little study in the literature discussing the applicability of different IBMs to yield-pseudoplastic flows. In this paper, the numerical methodology in several versions of immersed boundary methods, including the Volume Penalization IBM (VP-IBM), Indirect Imposition of Discrete Forcing IBM (ID-IBM), and Direct Imposition of Discrete Forcing (DD-IBM), is described. The numerical implementation of the methods is first validated using Newtonian benchmark cases. Then the solvers are utilized for the simulation of yield-pseudoplastic cases with stationary or moving particles. The investigation shows that in the explicit forcing methods, like VP-IBM and ID-IBM, the forcing procedure is conducted using an intermediate solution over the entire computational domain. The dependence of viscosity to strain rate in non-Newtonian cases leads to an inevitable sharp change in viscosity near the solid surface, giving rise to stiffness of the intermediate solution in the interface region. This makes the explicit forcing approaches based on intermediate solution incompatible with the flows with non-Newtonian yield-pseudoplastic rheology. On the other hand, the implicit forcing used in DD-IBM directly implements the boundary conditions on the boundary cells, making this method free from the perturbations related to the highly viscous solid region. In this regard, the simulation with the Direct Imposition IBMs like ghost-cell shows a good agreement between the predicted results and the experimental measurements for the stationary/moving cases. Therefore, it is shown that the DD-IBM is a reliable approach to modelling non-Newtonian flows.